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Saturday, July 26, 2008

Missing Pages

From a book, a number of consecutive pages are missing. The sum of the page numbers of these pages is 9808.

The Question: Which pages are missing?


Answer: Let the number of missing pages be n and the first missing page p+1. Then the pages p+1 up to and including p+n are missing, and n times the average of the numbers of the missing pages must be equal to 9808:

n×( ((p+1)+(p+n))/2 ) = 9808

In other words:

n×(2×p+n+1)/2 = 2×2×2×2×613

So:

n×(2×p+n+1) = 2×2×2×2×2×613

One of the two terms n and 2×p+n+1 must be even, and the other one must be odd. Moreover, the term n must be smaller than the term 2×p+n+1. It follows that there are only two solutions:

* n=1 and 2×p+n+1=2×2×2×2×2×613, so n=1 and p=9808, so only page 9808 is missing.
* n=2×2×2×2×2 and 2×p+n+1=613, so n=32 and p=290, so the pages 291 up to and including 322 are missing.

Because it is asked which pages (plural) are missing, the solution is: the pages 291 up to and including 322 are missing.


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